Mathematical Preliminaries

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1 CHAPTER 2 Mathematical Preliminaries This book views nonlinear dynamical systems as mathematical systems; namely a set associated with mappings defined on that set. The following lectures employ a number of concepts in topology, real analysis, and modern algebra. This chapter reviews these mathematical concepts with the goal of establishing notational conventions and defining basic mathematical concepts that will be invoked. 1. Sets and Algebras This section reviews notational conventions regarding sets and algebras. We start with the fundamental notation used in axiomatic set theory and then define several algebras (groups, rings, and fields) that will appear throughout these lectures. Sets: A point is an axiomatic concept (i.e. a concept with no formal definition). A set is a collection of points. A point, x is an element of a set X if it is contained in X. We denote this as x 2 X. Given two sets X and Y, if x 2 Y, for all x 2 X, then we say X is a subset of Y. We denote this as X Y. Two sets are equal if X Y and Y X. We denote the equality of two sets as X = Y. If X and Y are two sets, then the union of these sets is the set X [ Y, which we formally define as X [ Y = {x : x 2 X or x 2 Y } The intersection of two sets is the set X \ Y and is defined formally as X \ Y = {x : x 2 X and x 2 Y } The empty set, ;, is the set with no elements. Two sets are said to be disjoint if and only if X \ Y = ;. Given two sets X and Y that are subsets of a larger set U, the complement of Y in X is X Y = {x 2 U : x 2 X and x/2 Y } The notation X c denotes the complement of X in U. An important property about sets is De Morgan s law; namely that for any two sets A, B 2 X that (A[B) c = A c \ B c. This law can also be extended to a countable collection of sets as stated and proven in the following theorem. 25

2 26 2. MATHEMATICAL PRELIMINARIES THEOREM 1. Let {E i } be a collection (finite or infinite) of sets E i, then " # c [ (24) E i = \ (Ei c ) i i Proof: Let A and B be the left and right members of equation (24), respectively. if x 2 A then x/2 S i E i and so x/2 E i for any i. This means, therefore, that x 2 T i Ec i and so A B. Conversely, if x 2 B, then x 2 Ei c for every i and so x/2 E i for any i. This means x/2 S i E i and so x 2 [ S i E i] c thereby establishing B A. Since we just proved A B and B A, this implies A = B and the proof is complete. } Let x and y be elements of set X. The ordered pair, (x, y), is the association of elements x and y. For any two non-empty sets X and Y, the Cartesian product, X Y, is the set of all ordered pairs (x, y) of elements x 2 X and y 2 Y. Given two non-empty sets X and Y,amapping or function, f, from X into Y is a subset of X Y such that for each x 2 X there exists a unique y 2 Y with (x, y) 2 f. The value that map f takes at point x 2 X is denoted as f(x). The image of x under f is y = f(x). For a map f : X! Y, the set X is called the domain and the range of f is denoted as f(x). Given a function f : X! Y, the set of all ordered pairs {(x, f(x))} is called the graph of f. For any subset E Y, the inverse image of Y under f (denoted as f 1 (E)) is defined as f 1 (E) ={x 2 X : f(x) 2 E} If Y = f(x) = range(f), then f is said to be surjective or onto. The map f is injective or one-to-one if f(x) =f(y) implies x = y for all x 2 X and y 2 Y. A map that is surjective and injective is said to be bijective. If f is bijective one can show there exists a bijection f 1 : Y! X such that f 1 (f(x)) = x for all x 2 X. This map is called the inverse of f. A binary relation on a non-empty set X is a non-empty set R of ordered pairs (x, y) 2 X X. If a pair (x, y) 2 R, then we write xry and we say that x stands in relation R to y. A relation may be characterized by the following properties. reflexive if and only if xrx for all x 2 X. symmetric if and only if xry and yrx for all x, y 2 X, transitive if and only if xry and yrz implies xrz for all x, y, z 2 X. A relation R is said to be an equivalence relation if it is reflexive, symmetric, and transitive. If x and y are equivalent with respect to an equivalence relation, R, we write x R y. If is an equivalence relation for set X, then X can be partitioned into a collection of mutually disjoint sets called equivalence classes. Consider an element x 2 X and let [x] denote the equivalence class containing x, then y 2 [x] if and only if x y. A transitive relation R is called a total order if for all x, y 2 X one of the following three conditions applies xry, yrx, or x = y

3 1. SETS AND ALGEBRAS 27 This is called the trichotomy rule. Such order relations are often denoted using the less than sign, <. We say x apple y if either x<yor x = y. We say x>yif y<x. A set is said to be ordered if there is an order relation defined for all x 2 X. A transitive relation R is called a partial order if it is reflexive (x apple x for all x 2 X) and antisymmetric (x apple y and y apple x implies x = y). Note that under a partial order, not all pairs of elements in X stand in relation to each other. If X is a totally ordered set and E X, then if there exists 2 X such that x apple for all x 2 E, then we say is an upper bound for E. Suppose E X is bounded above by 2 X. If we know for any 2 X, where <, that is not an upper bound for E, then we say is the least upper bound (lub) or supremum of E. The supremum of E is often denoted as sup(e). In a similar way one defines a lower bound of set E with respect to order relation, >. is then a lower bound for set E if x for all x 2 X. The element is said to be a greatest lower bound (glb) or infimum of E if for all >, is not a lower bound. One often denotes the infimum of E as inf(e). A set X along with a collection of subsets of X is said to be a topology if the The trivial subsets, X and the empty set ;, are in. If A, B 2 then A \ B 2. If A, B 2 then A [ B 2. Given a set X with topology, the ordered pair (X, ) is called a topological space. Given a nonempty set X, abinary operator is a map f from X X into X. It is customary to associate a symbol such as + or with such operators. For instance if + is associated with f, then the value of f(x, y) is denoted as x + y. A binary operator + is said to be closed in X if x + y 2 X for all x, y 2 X. commutative if x + y = y + x for all x, y 2 X. associative if (x + y)+z = x +(y + z) for all x, y, z 2 X. Let + be a binary operator on a non-empty set X. An element e 2 X is called the identity of + if x + e = e + x = x for all x 2 X. If e is the +-identity over set X and if there exists x 1 2 X such that x + x 1 = e, then x 1 is called the right inverse of x. A similar left inverse can also be defined. If the left and right inverses are equal then we simply call x 1 the inverse of x. Algebras: The association of a set X and a collection of binary operators is called an algebra. Groups, rings, fields, and linear algebras represent special algebras of interest to us. Given a set X and a binary operator + defined on X, the ordered pair, G =(X, +), is called a group if

4 28 2. MATHEMATICAL PRELIMINARIES Closure: For any x, y 2 G, then x + y 2 G. Associativity: For all x, y, z 2 G, then (x + y)+z = x +(y + z). Identity: There is an identity element 0 2 G such that 0+x = x + 0=xfor any x 2 G. Inverse: For each x 2 G, there is an element x 2 G such that x +( x) =( x)+x = 0. Note that if the binary operator is also commutative (i.e. x + y = y + x for all x, y 2 G), then we say the group is abelian. Asubgroup is a subset S X of group elements is also a group. Given a set X and two binary operators, + (addition) and (multiplication), defined on X, the triple, R =(X, +, ) is called a ring if (X, +) is an abelian group with additive identity 0 2 R. The multiplication operator,, is associative in X. The multiplication and addition operators satisfy the distributive laws, x (y + z) =(x y)+(x z), (x + y) z =(x z)+(y z) A subring of ring R is a subgroup of R that is closed under multiplication. An ideal, I is a subset of elements of ring R that form an additive group that has the property than whenever x 2 R and y 2 I, then x y and y x belong to I. Given a set X and two binary operators, + (addition) and (multiplication), defined on X, the triple, F =(X, +, ) is called a field if (X, +, ) is a ring Multiplication is commutative and closed in X. There exists a multiplicative identity, 1 2 X, such that 1 6= 0 and every nonzero element of X has a multiplicative inverse denoted as x 1 or 1/x. 2. Linear Algebra Even though we will be dealing with nonlinear systems, the algebra of linear spaces will be useful to us. This is because all nonlinear systems may be viewed as maps between linear spaces of signals. This section reviews notational conventions regarding linear spaces and their use in defining signal spaces. We then define linear transformations and show that they too form a linear space. Linear transformations are useful when the input and output spaces are spaces of time-domain signals. In this case the linear transformation may be viewed as a linear dynamical system. Linear Spaces: Given a set X and field F, points of X will be called vectors. Points in F are called scalars. Consider a binary operation over X called vector addition. Consider a binary operation called vector dilation that maps ordered pairs in F X onto X. The 4-tuple L(X) =(X, F, +, ) is called a linear space if

5 2. LINEAR ALGEBRA 29 (X, +) is an abelian group. For any x 2 X and, 2 F there exists a vector x 2 X and x 2 X such that ( x) = ( x) = ( x) =( ) x. For any x 2 X, 1 x = x where 1 is the multiplicative identify of F. The vector addition and dilation are distributive ( + ) x = ( x)+( x) (x + y) = ( x)+( y) The standard example for a linear space occurs when F = R and X = R n. In this case addition, +, is standard vector addition and is scalar-vector multiplication. This linear space is sometimes called Euclidean space and it is simply denoted as R n. 2 In Euclidean space, the vectors are of the form, x = 6 4 x 1 x 2. x n 3. One may think of this as a function x : 7 5 {1, 2,...,n}!R. The graph of this function is shown in Fig. 1. You can imagine this graph as a sampled version of a continuous function (shown by the dashed line in Fig. 1). So one may think of function x : R! R n as an infinite dimensional vector. When the domain of x is time, then we will think of x as a time-domain signal. x(t) or x k x= x 1 x 2 x n and the function x (t ) FIGURE 1. Viewing a signal as a vector t Consider the set X = C(R, R n ) of continuous functions x : R! R n and let F be the real field. Define the signal addition operator in a component-wise manner, (x + y)(t) = x(t)+ y(t) for all t 2 R and x, y 2 X. Define the signal-dilation operator as ( x)(t) = x(t) for all t 2 R. For these operators, the right hand side refer to addition or dilation over the usual Euclidean vector space R n, whereas the left hand side operators are the signal versions of the operator. This means that signal addition and dilation inherit many of the axiomatic properties for a lienar space from the Euclidean vector space. It is trivial to conclude that these signal operators are commutative, associative, and distributive.

6 30 2. MATHEMATICAL PRELIMINARIES The only axiomatic property that does not directly follow from R n is closure. For signal addition to be closed, one must show that the sum (or dilation) of any two continuous functions is again continuous. This is clearly true, but it will have to follow from the definition of function continuity. Since these binary operators satisfy all of the axiomatic properties of a linear space, we can conclude that C(R, R n ) also has the algebraic structure of a linear space. Linear spaces are useful because they provide a useful framework for the modeling and analysis of signals and systems. We now review some basic facts about linear spaces. Let X be a linear space over a field F and let x 1,...,x n 2 X and a 1,...,a n 2 F. The vector P n i=1 a ix i is an element of X and is called a linear combination of vectors x 1,...,x n. A non-empty subset M of X is called a subspace of X is for any pairs of, 2 F and vectors x, y 2 M, we know x + y 2 M. In other words, M is closed under linear combination of its elements. For any linear space X, it should be apparent that X and {0} are also subspaces. Note that every subspace must contain the zero vector 0. If M and N are two subspaces, the direct sum of these spaces is denoted as M linear combinations of elements in M and N. N and it consists of all vectors formed from Consider a collection S of vectors from linear space X. The set of all linear combinations formed from elements of S also forms a subspace called the span of S. Given the collection S, if there exists a set of scalars, not all zero, such that the linear combination P n i=1 ix i =0, then one says the collection is linearly dependent. S is linearly independent if the a i s are all zero. Given a collection S = {x 1,...,x n } of vectors in X, we say that S forms a basis for X if S spans X or S is linearly independent. X is said to be finite dimensional if there exists a basis for X having a finite number of elements. It is known that the basis for finite dimensional linear spaces all have the same number of elements. For such spaces, the number of vectors in the basis is called the dimension of the space and is denoted as dim(s). Let X be a finite dimensional linear space of dimension n and let B x = {e 1,...,e n } be a basis for X. Let us consider a vector x 2 X and assume that it has two different linear combinations, nx nx x = i e i = i=1 i=1 ie i The difference between these two representations is nx 0= ( i i )e i i=1 The fact, however, that e i is a basis and hence linearly independent implies that i = i for all i =1, 2,...,n. One can therefore conclude that every nonzero vector x 2 X has a unique representation as a linear combination of a chosen basis set. The uniqueness of this representation prompts one to use the coefficients i as a concrete representation of the point x 2 X. These scalars are called coordinates of x with respect to the

7 2. LINEAR ALGEBRA 31 basis set B. One often arranges these coefficients as a column vector, [x] Bx 2 R n, as shown below 2 3 [x] Bx = n 2 R n 7 5 We can therefore conclude that the elements of any finite dimensional linear space can be represented by vectors in R n. Linear Transformation: We will think of linear spaces of functions as signal spaces. Asystem is then seen as a map between two signal spaces. When these signal spaces are time-domain signals, then the system is a dynamical system. The first type of system we will want to define are those defined by a linear transformation between two signal spaces. Let X and Y be two linear spaces over the field F. Consider a mapping G : X! Y and let G[x] denote the value that G takes for any x 2 X. This mapping is a linear transformation if G[( x)+( y)] = G[x]+ G[y] for all x, y 2 X and all, 2 F. We define the zero and identity transformations by the equation 0x = 0 and Ix = x for all x 2 X. These two transformations are clearly linear. We denote the set of all linear transformations from X into Y as L(X, Y ). A given linear transformation G : X! Y induces two special subspaces over X and Y. These subspaces are the kernel or null space and the image or range space of G. The null space of G is a subspace of X defined by ker(g) = x : x 2 X, Gx = 0 The image or range space of G is a subspace defined over Y Im(G) ={y 2 Y : y = Gx, x 2 x} Define the addition of two linear transformations, G, H 2 L(X, Y ) by (G + H)[x] =G[x]+H[x]. Define the dilation of a linear operator G 2 L(X, Y ) by the operator G that takes values ( G)[x] = (G[x]). We can again show that G + H and G are linear transformations and so L(X, Y ) is closed under these binary operations. Note that the definition of these two binary operations was done with respect to the binary operations defined in linear spaces X and Y, so again L(X, Y ) inherits the axiomatic properties regarding the commutativity, associativity, and distributivity of these operators. We can, therefore, conclude that L(X, Y ) is also a linear space. Since L(X, Y ) is a linear space, it has a basis. Let us determine a basis for an element G 2 L(X, Y ). Let B x = {e 1,...,e n } be a basis for X and let B y = {f 1,...,f m } be a basis for Y. Define a set of mn linear

8 32 2. MATHEMATICAL PRELIMINARIES transformations, E ij : X! Y for i =1,...,mand j =1,...,nsuch that E ij (e k )= jk f i where jk is the Kronecker delta. It can be shown that the set of linear transformations {E ij } form a basis for L(X, Y ). So any G 2 L(X, Y ) can be expressed as a unique linear combination of elements in this basis. In other words, there exist a unique set of { ij } in the field over which X and Y are defined so that G = mx nx i=1 j=1 Let us now consider the action that G has an x =2 X. The vector x has a unique representation, x = P n j=1 je j, with respect to the basis B x for X. If we consider the action of G on one of these basis vectors, e k, we can show it can be written as G[e k ]= mx i=1 ije ij with respect to the basis B y for Y. This means that the action of G on x can be written as 2 3 nx nx X m G[x] = G 4 j e j 5 = j ijf i = j=1 0 nx i=1 j=1 ij j ikf i 1 j=1 A f i i=1 This shows that the coordinates for the vector G[x] are [G[x]] By = 6 4 P n i 1j j 7 5 = n P n 1. mj j. m1 mn.. n = [G] Bx B y [x] Bx The linear operator, G, therefore has a concrete representation as a matrix, [G] Bx B y and so we can use matrix representations to more compactly represent these linear operators. You may view the matrix [G] Bx B y as a coordinate transformation between vectors defined with respect to basis B x and B y. 3. Norms for Signal and System Spaces A linear space L =(X, F, +, ) is an algebra that uses the binary operations of vector addition and dilation to characterize the relationship between elements of the set X defined on the field F. This algebra, however, has no notion of distance or length. Length is a topological concept and we will want to introduce a topology on top of our algebra if we wish to discuss the behavior of dynamical systems in a quantitative manner. There are many ways of introducing such a topological structure on top of the set X. The most general way is to simply define a collection,, of subsets of X that cover the original set X and that is

9 3. NORMS FOR SIGNAL AND SYSTEM SPACES 33 closed under set union and intersection. Such a collection is called a topology for X. We will adopt a more restrictive notion of distance that is based the concept of a norm. Norm: Consider a linear space, X, over field F. We define the norm of a vector x 2 X as any real number kxk 2R such that kxk 0 for all x 2 X kxk =0if and only if x = 0 k xk = kxk for all 2 F and x 2 X where is the absolute value for 2 F. kx + yk apple kxk + kyk for all x, y 2 X (triangle inequality). Given a linear space, X and a norm k k, then the mathematical system (X, k k) is called a normed linear space. When the norm is already understood from the context of the discussion, then we simply refer to X as a normed linear space. We will find it convenient to introduce norms for linear signal spaces formed from functions x : R! R n that are measurable (integrable). In particular, we define the L p norm for a signal x : R! R n as Z 1 kxk Lp = x(t) p dt 1 where p is any positive integer. With regard to the L p norm we can then define the L p normed linear space as 1/p L p = x : R! R n : kxk Lp < 1 In other words, L p consists of all real-valued functions whose L p norm is finite. There are three important subclasses of these spaces that occur when p =1, p =2, and p = 1. When p =1we end up with the space of absolutely integrable functions. When p =2we end up with the space of functions with finite energy. We define the L 1 norm of a function as kxk L1 = lim kxk L p =esssup x(t) p!1 In this case, the L 1 space consists of all signals with finite amplitude. Induced Gain: Since the set of linear transformations, L(X, Y ), is a linear space it is natural to ask whether one can place a norm on it. Recall that L(X, Y ) is the set of of linear transformations between two linear spaces X and Y. Assume that X and Y are normed linear spaces with norms k k x and k k y, respectively. Then it seems natural to use these signal norms to induce a norm on L(X, Y ). In particular, we say a linear operator G 2 L(X, Y ) is bounded if there exists M>0 such that t kg[x]k y apple Mkxk x for any x 2 X and y 2 Y. For any linear operator, we can then define the induced gain of G 2 L(X, Y ) as (25) kg[x]k y kgk sup = sup kg[x]k x6=0 kxk y x kxk x=1

10 34 2. MATHEMATICAL PRELIMINARIES Note that kgk 0 and kgk =0if and only if G equals the zero operator 0[ ]. Moreover, one can readily establish that the triangle inequality holds. So for any G, H 2 L(X, Y ), we can see that k(g + H)[x]k y kg + Hk = sup x6=0 kxkx apple kg[x]k y kh[x]k y sup +sup x6=0 kxk x x6=0 kxk x = G + H kg[x]+h[x]k y =sup x6=0 kxk x which establishes the triangle inequality for the induced gain. Since the induced gain satisfies all of the axioms required of a norm, we can conclude that the induced gain k kcan be used to transform L(X, Y ) into a normed linear space. An informal way of thinking about the induced gain kgk for a system is shown in Fig. 2. In this figure each plotted point is an ordered pair of the form, (kxk x, kg[x]k y ). The entire set of such points form a cloud of pairs that can be bounded above by a function of the form Lkxk x where L is some non-negative scalar constant. These overbounding curves are shown in the the figure. The smallest slope for this family of overbounding curves is what we called the induced gain of the system. An alternative way of defining the induced gain can therefore be written as (26) kgk =inf{l : kg[x]k y apple Lkxk x } We use the term induced for this norm because it is induced by the choice of norms for the system s input/output spaces. This is rather important for it means that whether or not an operator is actually bounded may depend on the choice of norm. For one choice, the map may have a finite induced gain, and yet for another choice of signal norm the induced gain may be unbounded. This will be important to us later when we consider stability concepts for input output systems that are based on the signal norms for the system s input space and output space. In some cases, it is possible to obtain explicit formulae for a system s induced gain. This is usually the case for linear dynamical systems. The following discussion shows how we can determine the induced gain for an LTI system when the input and output signal spaces are equipped with an L 1 norm. G[x] y L greatest lower bound on all slopes Consider a scalar and causal linear time-invariant system G whose input/output spaces are equipped with the L 1 norm. Assume that G has an impulse response g : R! R. Since this is a causal system, we know g(t) =0for t<0. Now consider an input signal x 2L 1. The system s outputs signal y : R! R may then be expressed as a convolution integral of x with the impulse response function g, x FIGURE 2. Induced Gain x y(t) = Z 1 1 g(t )x( )d

11 3. NORMS FOR SIGNAL AND SYSTEM SPACES 35 We re interested in determining the induced gain for this system with respect to the L 1 norm on both the input and output signals. Since the L 1 norm measures signal amplitude, let us consider the output signal s amplitude at time t 2 R. y(t) = apple apple apple = Z 1 1 Z 1 1 Z 1 1 g( )x(t g( )x(t g( ) x(t applez 1 g( ) d 1 applez 1 )d ) d ) d max x(t) t g( ) d kxk L1 = kgk L1 kxk L1 1 Since this inequality must hold for all t 2 R, one can readily conclude that kyk L1 = max y(t) applekgk L1 kxk L1 t This means that the L 1 -norm of the impulse response function g is also an upper bound on the L 1 -induced gain of the system G. In other words, kgk applekgk L1. We can show that kgk L1 is actually equal to the L 1 induced gain. This is a useful fact because it provides a formula that can be used in determining a quantitative value for the induced gain. To show equality, let us note that the bound implies that for any x 2L 1 that kg[x]k L1 kxk L1 applekgk L1 If, however, we can find a single x 2L 1 such that equality holds, in other words, (27) kg[x ]k L1 kx =kgk L1 k L1 Then we can deduce that kgk L1 = kg[x ]k L1 kx k L1 apple sup x kg[x]k L1 kxk L1 = kgk which implies that kgk L1 applekgk. So kgk L1 is also a lower bound on the induced gain. Coupling this with the fact that we also showed kgk L1 is an upper bound on kgk, we can conclude that kgk L1 = kgk. So if we could find an input signal x such that the equality in equation (27) holds, then we could conclude that the L 1 norm of the impulse response function, g, equals the L 1 induced gain of the operator G. Finding such an x, however, requires some insight into the operator. following input signal, In particular, let us consider the x(t ) = sgn(g( ))

12 36 2. MATHEMATICAL PRELIMINARIES This signal clearly satisfies kxk L1 =1and so using the convolution integral yields y(t) = = Z 1 1 Z 1 1 sgn(g( ))g( )d g( ) d = kgk L1 So here is a specific L 1 input signal for which the equality in equation (27) and so indeed, as claimed kgk = kgk L1. 4. Elementary Topological Concepts: limit points, openness, and closed sets: There are many ways of imposing a topological structure onto a set. One may do this with a collection of open sets (topology), a metric (distance between two elements), inner product (angle between two elements), or norm (length of an element). In our work we ll focus on using norms to impose a topological structure on a linear space. As discussed above, such linear spaces can be used to characterize systems and the signals these systems generate. When these signals are time-domain signals, then the associated systems are, of course, dynamic since their outputs are changing over time. With the introduction of such a topological structure, however, it will be convenient to define a number of topological concepts such as open sets, closure, compactness, and continuity. To some extent, all of these concepts have something to do with how the elements of a set are related when the size of that set becomes infinite. In particular, for a dynamical system that generates a state trajectory x : R! X mapping time to elements of normed linear space, we re interested in how the infinite nature of that trajectory constrains how that trajectory changes over time. These topological concepts will be very useful in characterizing those constraints. We start with the following set of fundamental definitions. Let X be a normed linear space with norm k k over a known field F. The first concept of interest to us will be a neighborhood. In particular the -neighborhood of p 2 X is the set N (p) ={x 2 X : kx pk < } We use this notion of a neighborhood to introduce the concepts of limit points, open, and closed sets in X. In particular a point p 2 X is a limit point of set S X if every -neighborhood of p contains a point q 2 S that is not equal to x. A set S X is closed if it contains all of its limit points. A set S X is open if for any p 2 S, there exists an -neighborhood such that N (p) S. A point for which this holds is called an interior point of S. So a set is open if all of its elements are interior points. The open and closed sets are fundamental topological constructions that will be used throughout our work. It is important to establish some fundamental results characterizing the relationship between these concepts. In particular, we first note that every neighborhood is an open set THEOREM 2. Every neighborhood N (x) for x in a normed linear space X is an open set.

13 4. ELEMENTARY TOPOLOGICAL CONCEPTS: LIMIT POINTS, OPENNESS, AND CLOSED SETS: 37 Proof: Consider a neighborhood N (x) in X and let y be any point of N (x). Then there is a positive real number h such that kx yk = h For all such points z such that ky zk <h, we then have kx zk applekx yk + ky zk < h + h = so that z 2 N (x). This means z is an interior point of N (x). Since the choice of y and z were arbitrary this means that all points of N (x) are interior points and so the neighborhood is open. } We can think of neighborhoods as canonical examples of open sets. Limit points of a set E X, on the other hand, represent points where elements E tend to cluster together. Essentially, this clustering means that in any arbitrarily small neighborhood about a limit point, one should be able to find infinitely many points of the set E. THEOREM 3. Let X be a normed linear space and let x be a limit point of a set E X. Then every neighborhood of x contains infinitely many points of E. Proof: This is proven by contradiction. Suppose there is a neighborhood N (x) that contains only a finite number of points of E. Let y 1,...,y n be those points of N (x) \ E that are distinct from x. Since this is a finite set, there exists a real number r>0 such that r = min kx y mk > 0. 1applemapplen The neighborhood N r (x), however, contains no point y of E such that y 6= x so that x is not a limit point of E. This contradiction establishes the theorem. } An open set E X may be seen as a set whose points are always surrounded by points of E. A closed set, on the other hand consists of limit points of the set E, namely those points around which points of E are clustering. It is not necessary, of course, that these clustering points actually be in E. A good example of such a situation occurs in the set of rational numbers (i.e. all real numbers that can be represented by a ratio of integers). If we look at any irrational number such as e or, one can always find a ratio arbitrarily close to the irrational number, but the number itself cannot be represented as a ratio. The relationship between open and closed sets can be described as complementary as seen in the following theorem, THEOREM 4. Let X be a normed linear space and let E X. This subset, E, is open if and only if its complement, E c, is closed. Proof: First suppose E c is closed and select a point x 2 E. Then x/2 E c and x is not a limit point of E c. This means there exists a neighborhood, N (x), such that E c \ N (x) =;. In other words, N (x) E and so x is an interior point of E. Since the choice of x 2 E was arbitrary, this means every point of E is an interior point and so E is open.

14 38 2. MATHEMATICAL PRELIMINARIES Conversely, suppose E is open. Let x be a limit point of E c, then every neighborhood of x contains a point of E c so that x is not an interior point of E. Since E is open, this means that x 2 E c. Again the choice of x was arbitrary in that it was any limit point of E c, so this means E c contains all of its limit points and so E c is closed. } One question of interest is whether the properties of open-ness or closure are preserved under standard set operations such as set union and intersection. The short answer is they are preserved for finite intersections and unions. But there is no guarantee that open-ness is preserved under countable intersections or that closure is preserved under countable unions. To formalize this result, we first need to establish the following THEOREM 5. The following statements are true for collections of open and closed sets. (1) For any collection {G i } of open sets, S i G i is open. (2) For any collection {F i } of closed sets, S i F i is closed. (3) For any finite collection G 1,...,G n of open sets T n i=1 G i is open (4) For any finite collection F 1,...,F n of closed sets S n i=1 F i is closed. Proof: Let G = S i G i. If x 2 G, then x 2 G i for some i. Since x is an interior point of G i, x is also an interior point of G and so G is open. This establishes statement 1. From theorem 1, we know (28) " # c [ F i = \ i i (F c i ) and by theorem 4 we known Fi c is open for all i. Therefore statement 1 implies that the set in (28) is open and so T i F i is closed. This establishes statement 2. Next let H = S n i=1 G i. For any x 2 H, there exists neighborhood N i of x with radius r i, such that N i G i for i =1, 2,...,n. Let r =min{r 1,...,r n } and let N be the neighborhood of radius r. Then N G i for i =1, 2...,nand so N H which implies H is open, thereby proving statement 3. Statement 4 follows from statement 3 by taking complements. } Any open set, E, can be turned into a closed set by simply augmenting it with its limit points. We call this the closure of E, denoted as E. The following theorem makes this more precise, by establishing that a set is equal to its closure if and only if E was closed in the first place. THEOREM 6. If X is a normed linear space and E X, then E is closed. and E = E if and only if E is closed.

15 5. SEQUENCES 39 Proof: If x 2 X and x /2 E, then x is neither a point of E nor a limit point of E. This means x has a neighborhood that does not intersect E. The complement of E is therefore open and so E is closed. Next if E = E, then the first assertion in the theorem implies that E is closed. Conversely, if E is closed then E E and so E = E. } One important example of this occurs when we consider the sup or inf of a set E. Clearly the sup(e) and inf(e) are a limit point of E by virtue of their definition. So the sup (inf) can only be contained in E if E is already closed. This fact is stated and proven below. This is an important clue that will be used later to ensure the existence of solutions to optimization problems. THEOREM 7. Let E be a non-empty set of real numbers that is bounded above. Let y =supe, then y 2 E and hence y 2 E if E is closed. Proof: If y 2 E, then y 2 E. Assume y /2 E. For every h>0 there exists a point x 2 E such that x h<x<y, for otherwise y h would be an upper bound of E. Therefore y is a limit point of E and so y 2 E. } 5. Sequences A sequence is a function x : Z! X from the set of integers, Z, onto a set X. In our case, we ll take X to be a normed linear space. We denote a sequence {x i } i2i where I Z is a set of indices. In general, our sequences are infinite in the sense that I = Z and so we denote such sequences as {x i } 1 i=1. A sequence {x i } 1 i=1 is convergent in normed linear space X if there exists a point x 2 X such that for all >0there is an integer N>0such that for any n N, we can show kx n xk <. If {x i } is convergent then the point x is called the limit point of the sequence and we write x n! x or lim n!1 x n = x. A sequence that has no limit point is said to be divergent. Consider a sequence of functions {f n (t)} 1 n=0 where f n (t) =t n for 0 apple t apple 1. In the L 1 signal space (i.e. all bounded functions), this sequence is convergent to the function ( 0 0 apple t<1 f(t) = 1 t =1 which is also a bounded function. If we consider this sequence to be in the linear space of continuous functions C[0, 1] (note that each f n is a continuous function on [0, 1]) then this sequence is divergent because the limiting function is discontinuous and hence not in C[0, 1]. Convergent sequences are useful tools in the analysis of nonlinear systems. But the definition as stated above is difficult to verify because one needs to know what the limit point is, x. We therefore introduce new notion of a sequence known as a Cauchy sequence. A sequence {x i } in normed linear space X is a Cauchy sequence if for every >0there is an integer N such that kx n x m k < for all n, m N.

16 40 2. MATHEMATICAL PRELIMINARIES Let E be a nonempty subset of the normed linear space X and let S be the set of all real numbers of the form kx yk with x, y 2 E, S = {r 2 R : r = kx yk,x,y 2 E} Then we define the diameter of S as diam(e) =sup(s). If {x n } is a sequence in X and if E N consists of points {p N+i } 1 i=0, then clearly {x n} is a Cauchy sequence if and only if (29) lim diam(e N )=0 N!1 THEOREM 8. In any normed linear space, every convergent sequence is a Cauchy sequence Proof: If x i! x then for any >0there is an integer N such that kx x n k < for any n N. Hence kx n x m kapplekx n xk + kx x m k < 2 for n, m N and so {x n } is Cauchy. } In general not all Cauchy sequences are convergent (see above example). Normed linear spaces in which every Cauchy sequence is convergent are said to be complete. Complete normed linear spaces are sometimes called Banach spaces. All Euclidean spaces, R n, are complete as are the functions spaces, L p [Roy68]. We will not establish the completeness of L p spaces, but it will be useful to establish that R n is complete. To do this, however, requires we introduce a new topological concept; compactness. 6. Compactness Compactness is an important concept especially with regard to the existence of extreme points for continuous functions. Its importance is based on the observation that the behavior of finite sets and infinite sets is very different. The following discussion of compactness and its role in mathematical analysis is based on [Hew60]. Each of the following statements, for example, is true when a set X is finite, All functions are bounded: If f : X! R is a real valued function on X then there exists M>0 such that f(x) apple M for all x 2 X. All functions attain a maximum: If f : X! R is a real valued function on X, there must exist at least one point x 2 X such that f(x ) f(x) for all x 2 X. All sequences have constant subsequences: If {x i } i2i is a sequence in X then there must exist a subsequence {x ij } ij2i that is constant for some c 2 X. All covers have finite subcovers: If {V i } i2i are subsets of X that cover X (i.e. X [ i2i V i ), then there exist a finite set J I of sub-indices such that the finite sub-collection {V ij } ij2j still covers X (i.e. X [ ij2jv ij ). The first statement - all functions on a finite set are bounded - is an example of a local-to-global principle. Namely that the assertion of local boundedness by a constant M that is a function of x implies a global bound for all x 2 X by a constant that is independent of x. This is true for a finite set because one can

17 6. COMPACTNESS 41 enumerate the set of elements and take the largest as the bound. This is, in general, not true for infinite sets. The second statement - all functions have a maximum - is an example of the maximum principle. It clearly holds since we can again enumerate the elements of X to find the maximum. In an infinite set the maximum is a limit point and if X is open then it may not contain all of its limit points. The third statement - every sequence has a constant subsequence - is sometimes called the infinite pigeonhole principle. Since the sequence only cycles through a finite number of elements, it obviously allows us to identify a constant subsequence. This will not, in general be true for sequences defined on infinite sets. The last statement - every cover has a finite sub-cover - is what we often define as countable compactness and again may not be true for arbitrary infinite sets. When we endow our set X with some topological structure, like a norm or metric, then some objects exhibit properties similar to those above that are enjoyed by finite sets. For example consider X =[0, 1], the closed unit interval. In general, the above four properties are all false for this X. But if we modify each assertion by inserting a topological concept such as continuity, convergence, or open-ness, then we can establish these assertions for [0, 1]. All continuous functions are bounded on X =[0, 1]: All continuous functions achieve a maximum on X =[0, 1]: All sequences in [0, 1] have convergent subsequences: All open covers have finite subcovers: In contrast all four of these statements continue to be false for sets like the open unit interval, (0, 1) or the real line, R. Compactness, therefore, is a powerful property of spaces and it is used in many ways. One may use the local-to-global principle to establish some local constraints on a function that can then be used to infer a global constraint. Another use is to establish conditions to locate function extrema; this is done for instance in optimal control. A third is to partially recover the notion of a limit when dealing with non-convergent sequences, this is what is done in characterizing limit cycles. In our studies we will need compactness in a variety of places. So this subsection formally defines what it means for a set to be compact and then states and proves various theorems about such sets. Let {G i } i2i be a collection of open subsets of X where I is a set of indices. This collection is called an open cover for E X if E S i2i G i. A subset K of a normed linear space X is said to be countably compact in X or just compact in X if every open cover of K contains a finite subcover. More explicitly this means that if we re given an open cover {G i } i2i of K where I is a set of indices (possibly infinite), then there are finitely many indices i 1,...,i n such that K G i1 [ G i2 [ [G in

18 42 2. MATHEMATICAL PRELIMINARIES There is a close relationship between compact subsets and closed sets. In fact every compact subset of a normed linear space is also closed. The converse relation, however, only holds if the closed set is taken from a set that is already known to be compact. This is reflected in the following two theorems. THEOREM 9. Compact subsets of normed linear spaces are closed. Proof: Let K be a compact subset of a normed linear space X. Suppose x 2 X and x/2 K. If y 2 K, let V y and W y be neighborhoods of x and y, respectively of radius less than 1 2 kx are finitely many points y 1,...,y n in K such that K W y1 [ [W yn = W yk. Since K is compact, there If V = V y1 \ \V yn, then V is a neighborhood of x that does not intersect W. So V K c so that x is an interior point of K c, thereby establishing that K c is an open set and so but K is closed. } THEOREM 10. Closed subsets of compact sets are compact. Proof: Suppose F K X with F closed in X and K compact. Let {V i } i2i be an open cover of F. If F c is added to the collection {V i } we obtain an open cover,, of K. Since K is compact, there is a finite subcollection of which covers K and hence F. If F c is a member of, we may remove it from and still retain an open cover of F, thereby showing that there is a finite subcollection of {V i } that covers F. } As mentioned above, an important aspect of compact sets is their support of the local-to-global principle and the maximum principle. The foundation for this support is established in the following theorem that asserts that any infinite set of a compact set K has its limit point in K. If that limit point is the sup of a function, then this theorem lays the foundation for establishing the existence of global bounds on a function s value. THEOREM 11. If E is an infinite subset of a compact set K, then E has a limit point in K. Proof: If no point of K were a limit point of E, then each y 2 K would have a neighborhood V y which contains at most one point of E (namely y). It is clear that no finite subcollection of {V y } can cover E; and the same is true of K, since E K. This contradicts to compactness of K. } When the compact space, X, is actually the Euclidean space, R k, it becomes possible to sharpen our notion of compactness and its relation to compact and bounded sets. To establish this relationship we first need to define what we mean by a bounded set. In particular, A set S X is bounded if there exists L>0 such that kx yk <Lfor all x, y 2 S. When we focus on R k it will also be useful to introduce a multidimensional notion of a closed interval; what is usually called a k-cell. Let X = R k, let a, b 2 R k, and let x =(x 1,...,x k ) denote a vector in R k.ak-cell I[a, b] consists of all points x 2 R k such that a i apple x i apple b i for i =1, 2,...,k. It can be proven that k-cell s are compact sets. With this fact, we can state the following theorem. The first two statements in this theorem are usually known as the Heine-Borel theorem. This theorem states that a subset of R k is compact if and only if it is closed and bounded. THEOREM 12. (Heine-Borel Theorem) The following three assertions are equivalent set for subset E R k.

19 7. SEQUENCES AND COMPACTNESS: 43 (1) E is closed and bounded. (2) E is compact (3) Every infinite subset of E has a limit point in E. Proof: If statement (1) holds, then E I[a, b] for some k-cell I[a, b] and 2 follows from that fact that all k-cells are compact. Theorem 11 implies that statement 2 implies statement 3, so it remains to show that statement 3 implies statement 1. If E is not bounded, then E contains points x n with x n >n, n=1, 2,... The set S consisting of these points x n is infinite and clearly has no limit point in R k, hence has none in E. So statement (3) implies that E is bounded. If E is not closed then there is a point x 0 2 R k that is a limit point of E but not a point of E. For n = 1, 2...,1 there are points x n 2 E such that x n x 0 < 1/n. Let S be the set of these points, then S is infinite. S has x 0 as a limit point and it has no other limit point in R k. For if y 2 R k and y 6= x 0, then x n y x 0 y x n x x 0 y n 2 x 0 y for all but finitely many n; this shows y is not a limit point of S. So S has no limit point in E and so E must be closed if statement 3 holds. } 7. Sequences and Compactness: One of the important features of compact sets is that Cauchy sequences are convergent in such sets. Recall that sequences provide a useful tool in studying dynamical systems, but that the formal definition of convergence assumes we know the limit point. This is why we introduced the notion of a Cauchy sequence since the clustering exhibited by such sequences is easier to verify. We now turn to examining why compactness ensures that Cauchy sequences are convergent. The following theorem provides a useful starting point by considering a sequence of compact sets whose diameter asymptotically goes to zero. For this sequence, we show that there is a limiting set and that limiting set is exactly one point in the space X. THEOREM 13. Let {K i } 1 i=0 be a sequence of compact sets in normed linear space X such that K i K i+1 for i =1, 2,...,1. If then T 1 i=1 K i consists of exactly one point in X. lim diam(k i)=0 i!1 Proof: Let K = T 1 1 K i. Assume it is empty but note that the intersection of any finite subcollection of K i must be nonempty. Fix a member K 1 of {K i } and let G i = K c i. Assume no point of K 1 belongs to every

20 44 2. MATHEMATICAL PRELIMINARIES K i. This means the sets {G i } form an open cover of K 1. Since K 1 is compact there are finitely many indices i 1,...,i n such that K 1 G i1 [ [G in. This means, however that K 1 \ K i1 \ \K in would be empty which would contradict the fact that any finite subcollection of K i s must be nonempty and so K must also be nonempty. So since K is not empty, if K contains more than one point, then diamk >0. But for each n, K K n, so that diam(k n ) diam(k), which contradicts the assumption that diam(k n )! 0. } The preceding theorem characterizes the convergence of a sequence of sets, but those sets can also be related back to a Cauchy sequence through the diameter condition in equation (29). This means, therefore, that the above theorem can be used to prove that any Cauchy sequence in a compact space is convergent to a point in X. THEOREM 14. If X is a compact normed linear space and if {x n } is a Cauchy sequence in X, then {x n } converges to some point of X. Proof: Let {x n } be a Cauchy sequence in compact space X. For N = 1, 2,...,1, let E N be the set consisting of points x N,x N+1,... Then lim diame N =0 N!1 Being a closed subset of the compact space X, each E N is compact by theorem 10. Also E N+1 E N so that E N+1 E N. Theorem 13 shows there is a unique x 2 X which lies in every E N. Let >0 be given. Since lim N diame N =0, there is an integer N 0 such that diam(e N ) < if N N 0. Since x 2 E N, it follows that kx kx x n k < for all n N 0 which is precisely x n! x. } yk < for every y 2 E N and so for all y 2 E N. In other words, The preceding theorems indicate a strong connection between convergent Cauchy sequences and compact spaces. In fact, an alternative characterization of compactness (sequential compactness) simply asserts that a set is sequentially compact if every sequence has a convergent sub-sequence. To make this more precise, let us first define what we mean by a subsequence. Given a sequence {p n }, consider a sequence {n k } of positive integers such that n 1 <n 2 <n 3 <. Then the sequence {p ni } is called a subsequence of {p n }. If {p ni } converges, its limit is called a subsequential limit of {p n }. Note that {p n } converges to p if and only if every subsequence of of {p n } converges to p. The following theorem asserts that any sequence in a compact space has a convergent subsequence and the restriction of this fact to R is usually called the Bolzano-Weierstrass theorem. The converse relation is also true for normed linear spaces, but it may not be true, in general, for any topological space. THEOREM 15. If {p n } is a sequence in a compact normed linear space, X, then some subsequence of {p n } converges to a point of X. (Bolzano-Weierstrass Theorem): When X = R k, every bounded sequence contains a convergent subsequence.

21 8. CONTINUOUS FUNCTIONS AND THE MAXIMUM PRINCIPLE 45 Proof: Let E be the range of {p n }. If E is finite, then there is a p 2 E and a sequence {n i } with n 1 <n 2 < n 3 <, such that p n1 = p n2 = = p. The subsequence {p ni } obtained clearly converges to p. Now let E be infinite. Since E is an infinite set on a compact set, theorem 11 implies that E has a limit point p 2 X. Choose n 1 so that kp p n1 k < 1. Having chosen n 1,...,n i 1 and since every neighborhood of p has an infinite number of points we can find an integer n i >n i 1 so that kp p ni k < 1/i. So this means {p ni } converges to p and so the first assertion in the theorem is established. If X = R k, we can then use the Heine Borel theorem to deduce that every bounded subset of X lies in a compact set and so by the first part of theorem we establishes the convergence of the subsequence. } The fact that every compact set in R k is also closed and bounded and that Cauchy sequences are convergent in compact spaces makes it easy to establish that R k is a complete normed linear space. The first result below is known as the Weierstrass theorem and follows immediately from theorem 11. The second theorem 17 proves every Cauchy sequence in R k is convergent by appealing to the Heine-Borel theorem and then invoking theorem 14. THEOREM 16. (Weierstrass Theorem) Every bounded infinite subset of R k has a limit point in R k. Proof: Being bounded, the set E is a subset of k-cell I R k. I is compact and so E has a limit point in I by theorem 11. } THEOREM 17. (R k is Complete) In R k, every Cauchy sequence converges. Proof: Let {x n } be a sequence in R k. Define E N as the set consisting of x N, x N+1,...,. For some N, E N < 1. The range of {x n } is the unition of E N and the finite set {x 1,...,x N 1 }. Hence {x N } is bounded. Since every bounded subset of R k has a compact closure in R k (Heine-Borel), we can use theorem 14 to conclude that the Cauchy sequence is convergent. } 8. Continuous Functions and the Maximum Principle One useful aspect of compactness appears when we consider continuous functions on compact spaces. In particular, continuous functions have maxima when they are defined over compact spaces. This is useful because it can be used to determine whether or not an optimization problem is well posed in the sense of a solution existing. In R k this is important, but it is more useful when we consider dynamic optimization problems found in optimal control where the space over which our function acts is a function space. In this case, establishing the existence of extreme points for the optimization problem is non-trivial and establishing the compactness of the associated input space is critical for ensuring the existence of an optimal control. Let us begin by defining the limit of a function, f. In particular, let X and Y be two normed linear spaces. Suppose E X and that there is a function f : E! Y. Let p be a limit point of E, then we write f(x)! q